I'd go with the second one instinctively, though I can't see why the first one wouldn't also work. This sort of conundrum stymies me all the time. Consider the following sentences:
Everyone ran for their lives.
Everyone ran for their life.
We all crossed our hearts and wished for luck.
We all crossed our heart and wished for luck.
I hope you all brought your wallets.
I hope you all brought your wallet.
In each pair, which is correct? Either one in each pair can lead to confusion depending on how technically you choose to read them. In general, I find either works.